Question

Suppose that $f_n: [0,1] \to \mathbb{R}$ are continuous function $\forall n \in \mathbb{N}$ and there exist $f: [0,1] \to \mathbb{R}$ such that $\forall x \in [0,1]$, $f_n(x) \to f(x)$ as $n \to \infty$. Then $f$ is continuous on $[0, 1]$.

Name: Suppose that fn: [0,1] →ℝ are continuous function ∀ n ∈ℕ and there exist f: [0,1] →ℝ such that ∀ x ∈ [0,1], fn(x) → f(x) as n →∞. Then f is continuous on [0, 1].
Uploaded: 2023-11-21T04:36:41-08:00
Duration: 6 min 11 s
Channel: Madhur L
Description: Suppose that fn: [0,1] →ℝ are continuous function ∀ n ∈ℕ and there exist f: [0,1] →ℝ such that ∀ x ∈ [0,1], fn(x) → f(x) as n →∞. Then f is continuous on [0, 1].

          Suppose that $f_n: [0,1] \to \mathbb{R}$ are continuous function $\forall n \in \mathbb{N}$ and there exist $f: [0,1] \to \mathbb{R}$ such that $\forall x \in [0,1]$, $f_n(x) \to f(x)$ as $n \to \infty$. Then $f$ is continuous on $[0, 1]$.

Suppose that fn: [0,1] →ℝ are continuous function ∀ n ∈ℕ and there exist f: [0,1] →ℝ such that ∀ x ∈ [0,1], fn(x) → f(x) as n →∞. Then f is continuous on [0, 1].

Added by Jennifer O.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

11/21/2023

Step 1

Step 1: We know that for each n, fn is continuous on [0, 1]. Show more…

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Either rigorously prove or provide a counterexample, please! (e) Suppose that fn : [0, 1] → R are continuous functions for all n ∈ N and there exists f : [0, 1] → R such that for all x ∈ [0, 1], fn(x) → f(x) as n → ∞. Then f is continuous on [0, 1].